The molecular structure of a synthetic polymer determines its end-use and its process characteristics, such as hardness, tensile strength, drawability, elastic modulus, and melt viscosity. The most important determinants that define the molecular structure of synthetic polymers are the chemical nature of their repeating units, their molecular weight distribution (MWD), and their molecular topology (branching).
The separation of a polymer sample by gel permeation chromatography (GPC) followed by detection allows one to determine the molecular weight distribution (MWD) and the intrinsic viscosity law (IVL) of a polymeric sample. The IVL can reveal the branching properties of a sample. Another term used for GPC is size exclusion chromatography (SEC).
In a GPC separation, a molecule's effective size, not its molecular weight per se, determines its elution volume. For example, two polymers may have the same molecular weight, but may differ in composition or branch topology. Such differences can lead to different elution volumes. Thus, elution volume by itself determines a sample's molecular weight distribution in a relative, not an absolute sense.
Following separation, one or more detectors record the physical properties of the eluent stream. It is the mathematical analysis of these data from which the MWD and IVL of the sample can be obtained.
When narrow standards of known molecular weight (MW), the sum of atomic weights of all atoms in a molecule, are available, and these standards are of the same polymeric material as the sample, molecular weight calibration is straightforward to perform. Injections of the narrow standards determine the chromatography column set's MW calibration curve (log MW versus elution volume). Under these circumstances, only a single concentration detector (such as a refractive index (RI) detector) is needed to determine the MWD.
If no such standards are available, dual detection schemes can be employed to determine the sample's MWD. A first typical dual detection system employs refractive index (RI) and light scattering (LS) detection. A second known system employs RI and viscometric (V) detection. The data from both such systems can be used to determine a sample's MWD.
The RI-V detection system allows the additional determination of the molecular topology of the polymer through a plot of its intrinsic viscosity law (IVL). The IVL of a sample is its intrinsic viscosity as a function of its molecular weight. The intrinsic viscosity is the ratio of the sample's specific viscosity (measured by the viscometer) to its concentration (measured by the concentration detector).
In the case of RI-V detection, polymer calibration standards are needed to determine the MWD and IVL, but the repeat units do not need to have the same chemical nature as those of the polymer sample. The RI-LS detection strategy requires no additional calibration standards.
For both systems, the RI detector is used to measure a peak's concentration profile. In place of an RI detector, a suitably calibrated UV/V is absorbance detector, evaporative light scattering detector (EVS), or infrared (IR) detector may be substituted to perform the same function.
The LS and V detectors are commonly referred to as "molecular-weight sensitive" detectors. Such detectors respond to the product of the sample's concentration and its molecular weight raised to some power.
A combination of a concentration detector and a molecular-weight-sensitive detector greatly increases the information content available from a typical analysis. But the increase in the available information expands the complexity of data analysis. For both systems, algorithmic methods are required to analyze the detector responses in order to obtain the sample's MWD. In the RI-V system, an additional algorithmic method is needed to determine the sample's IVL.
The accuracy and the precision of the MWD and the IVL depend not only on the quality of the data obtained from the respective chromatographic systems, but also on the details of the data analysis methodology. Thus, data analysis methods become an essential element in the GPC analysis of polymers.
At evenly spaced time intervals, each detector records a measurement of the properties of the separated sample as it elutes from the column and passes through a detector's flow cell. Each measurement, averaged over a narrow time range, corresponds to a narrow range in the sample's molecular weight distribution. The molecular weight range corresponding to a measurement recorded at a single time interval is referred to as a "slice".
A slice can be referenced by its slice number, its elution time, or its elution volume. Typically, elution volume is obtained by multiplying the elution time of a slice by the nominal flow rate of the pump. A particular slice is typically referred to by its slice number or elution volume, with no loss of generality.
The RI, LS, and V detectors are used to measure, respectively, the concentration ci, the Rayleigh ratio Ri, and the specific viscosity .eta.sp,i, for each slice i.
If a pair of chromatographic profiles are obtained from two detectors, the ratio of the respective responses can be formed to obtain useful quantities. For example, if ci and Ri are the measurements of concentration and the Rayleigh ratio from the ith slice, the ratio .rho.i=Ri/ci is proportional to the molecular weight of that slice. If ci and .eta.sp,i are the slice measurements of concentration and specific viscosity, the ratio [.eta.]i=.eta.sp,i/ci equals the intrinsic viscosity of the slice.
Both the MWD and IVL need to be measured over the whole peak region. Typically, this is accomplished by fitting a smooth parameterized model to these ratios as a function of slice number or elution volume. The logarithm of the ratios, log(Ri/ci) and log(.eta.sp,i/ci), both tend to be nearly linear functions of elution volume, so low-order polynomials as a function of elution volume are typically fit to these quantities.
A major problem is that the noise present in the detector responses introduces errors in the quantities computed from slice measurements. Each of the known detectors employed for GPC contains non-idealities in their responses. Typically, these non-idealities fall into two categories, baseline drift and stochastic detector noise. Detector noise can also be referred to as system noise. Baseline drift in a thermally stabilized chromatograph is accurately compensated for by baseline correction procedures.
Detector noise is an irreducible component of the measurement process. The origin of this noise, seen as fluctuations in the baseline, is the result of several fundamental phenomena. One is the shot noise of the light sources such as in RI and LS detectors. Other origins are thermal noises associated with amplifiers in all detectors; fluctuations in the pump flow rate; and thermal variations. Particulate, contaminants, and bubbles can also add additional noise components to the signal.
The net result of these effects is manifested in stochastic noise added to each slice measurement. Such additive noise has zero mean and a well-defined standard deviation. The standard deviation of the noise will in general be different for the different detectors, but each detector's noise is constant throughout the separation.
The effect of the detector-noise-induced error in the slice-measurements is to introduce error in the quantities log(Ri/ci) and log(.eta.sp,i/ci). Because ci is in the denominator, the noise in these quantities increases as the response in the concentration profile decreases. Because of the logarithm, the noise in these quantities also increases as the response of the molecular-weight-sensitive detector decreases. Thus, the noise in log(Ri/ci) and log(.eta.sp,i/ci) increases dramatically in the tails (leading and trailing edges) of a chromatographic peak.
Further, near the peak tails, the noise in the responses can cause the ratios to have negative values. The logarithm of a negative number is not defined. Eliminating such slice data will bias the results. This issue of dealing with the logarithm of the response ratios is well described in the recent paper "Systematic Deviations due to Random Noise Levels in Size Exclusion Chromatography Coupled With Multi Angle Laser Light Scattering" by P. Tackx and F. Fosscher, 1997, Anal. Comm. 34, 295-297.
The question remains as to how to fit a smooth model curve to log(Ri/ci) and log(.eta.sp,i/ci) across a complete peak profile given the presence of detector noise.
Prior to the present invention, it was assumed that in the peak tails, the quantities log(Ri/ci) and log(.eta.sp,i/ci), because they appeared dominated by noise, contained no useful information. Least-squares fitting of models were then confined to the "heart" of the peaks where the ratios' signal-to-noise ratio (SNR) were high. Users were required to manually decide the demarcation between the region to which the model is fit, and regions to exclude from the model fit.
However, to determine the MWD and IVL throughout the peak, the values of these quantities were nevertheless needed in the peak tails. The prior practice was to extrapolate the model results to the peak tails. The results obtained from such extrapolations are notoriously sensitive to the choice of the fitting region and the extrapolation method, as Tackx and Fosscher point out.